Matheus G. C. Cunha | Searcharxiv

arXiv2026

Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.

math.DS↗

arXiv2024

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a Shilnikov sliding connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that it is strictly greater than $0$ and strictly less than $1$, and its one-dimensional Lebesgue measure is 0. Moreover, we prove that the closure of the local invariant set is a Cantor set and has the same Hausdorff dimension and Lebesgue measure of the original invariant set. Furthermore, it is given by the invariant set adjoined with the set of all pre-images of the regular-fold point.

math.DS↗